Question: Simplify the following expression and state the condition under which the simplification is valid: $p = \dfrac{y^2 - 16y + 63}{y^2 - 4y - 45}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{y^2 - 16y + 63}{y^2 - 4y - 45} = \dfrac{(y - 7)(y - 9)}{(y + 5)(y - 9)} $ Notice that the term $(y - 9)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(y - 9)$ gives: $p = \dfrac{y - 7}{y + 5}$ Since we divided by $(y - 9)$, $y \neq 9$. $p = \dfrac{y - 7}{y + 5}; \space y \neq 9$